130 XIII INTEGRATION Assertion (i) is a consequence of (13.8.1) and (13.8.2), applied to the sequence ( — <pAn). Assertion (ii) follows from (13.8.2). The first part of (iii) 00 and the formula (13.8.7.2) follow from (13.8.2), the inequality <pA ^ Z <PAn n=l and (13.5.8). Finally, if the sets An are pairwise disjoint, then we have 00 <PA = Z ^An> anc* (13.8.7.3) follows from (13.8.5), applied to the functions <pAn. n=l Example (13.8.8) Let/be a regulated function on R with compact support (and therefore bounded, by (7.6.1)). Then/is integrable with respect to Lebesgue measure L If J = [tf, 6] is an interval containing Supp(/), we have (13.8.8.1) I/UI- f(t)dt in the sense of (8.7). For we have |/| ^ ||/|| <ps, and by (7.6.1), (8.7.8) and (13.8.4) it is enough to prove (13.8.8.1) for a step function with compact support (7.6). Such a function is a linear combination of characteristic functions of intervals, so that finally we are reduced to the case where/ = (pl, with I an interval; and in this case our assertion follows from (13.7.8). PROBLEMS 1. On the discrete space N, let/A be the measure for which ju({/z}) = 1 for all n. Show that the functions n\-^-un which arejit-integrable are those for which the series with general term un is absolutely convergent. They are also those for which the upper (or the lower) .. oo integral is finite. We have undfji(n)=^un, and the space ^R(N,U) is the Banach J n = 0 space I1 (Section 5.7, Problem 1). 2. (a) Show that if / and g are two real-valued functions i>0 defined on X, then ju,*(sup (/, g)) +/x*(inf (/, g)) ^ p.*(f) + /**(#. (b) Let/be a real-valued function on X such that /><-*(/) is finite. Show that there exists an integrable function/! ^/such that ftC/i) = ju,*(/). If/2 is another integrable function such that/2 ^/and /x(/2) = ^u.*(/), then/! and/2 are equivalent. (c) Let / be a real-valued function on X such that both ft*(/) and //,*(/) are finite. Let g, h be integrable functions such that g <=/<^ h and /x(#) = /x*(/), ju,(/z) = p*(f). Show that p*(f~ g) = p*(h -/) = 0 and thaten any e > 0, there exists a function u e «2rR(X) such that this defines an order relation on the set of equivalence classes (with